IAC-04-A.4.10

INTERNATIONAL SPACE STATION LEAKLOCALIZATION USING VENT TORQUE

ESTIMATIONJong-Woo Kim, John L. Crassidis

Department of Mechanical and Aerospace EngineeringUniversity at Buffalo, The State University of New York

Amherst, NY 14260-4400716-645-2593 ext. 2246

boungdong@yahoo.com, johnc@eng.buffalo.edu

Srinivas R. VadaliDepartment of Aerospace Engineering

Texas A&M UniversityCollege Station, TX 77843-3141

979-845-6051svadali@aero.tamu.edu

Adam L. DershowitzUnited Space Alliance

NASA Johnson Space Center, Code DF64Houston, TX 77058

281-483-5410dersh@alum.mit.edu

ABSTRACTA new method is presented in this paper to localize air leaks on the International Space Station based on the spacecraftattitude and rate behavior produced by a mass expulsion force of the leaking air. Thrust arising from the leak generates adisturbance torque, which is estimated using an unscented filter with a dynamical model (including external disturbancessuch as aerodynamic drag and gravity-gradient). The leak location can be found by estimating the moment arm of theestimated disturbance torque, assuming that leak is causedby only one hole. Knowledge of the leak thrust magnitudeand its resulting disturbance torque are needed to estimatethe moment arm. The leak thrust direction is assumed tobe perpendicular to the structure surface and its magnitudeis determined using a Kalman filter with a nozzle dynamicsmodel. There may be multiple leak locations for a given response, but the actual geometric structure of the space stationeliminates many of the possible solutions. Numerical results show that the leak localization method is very efficient whenused with the conventional sequential hatch closure or airflow induction sensor system. A user friendly computer code hasbeen developed to find the leak location with the proposed method.

INTRODUCTION

The International Space Station (ISS) is orbiting ina 51.6◦ inclination near-circular Low-Earth-Orbit (LEO)with an altitude between 370 and 460 km, and is expectedto have a minimum operational lifetime of 15 years. Be-cause of the large structure, long lifetime and orbit char-acteristics,1 the ISS may be subject to impacts of hypervelocity particles such as micro-meteorites and space de-

bris that can severely damage the station. This damagemay threaten the safety of the crew if the pressurized wallof a module is perforated, which may result in significantair loss. Collisions with other objects are another possiblecause of a leak, as occurred in the Russian Space StationMir in 1997. To protect the ISS from the impact damages,various debris shields have been designed. Heavy shieldsare placed in the forward facing area which is likely to be

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hit frequently, and fewer shields are used in the nadir-facingand aft area.2

Perforations in a pressurized module will result in a rapidtemperature and pressure decrease. Therefore fast determi-nation of the extent and location of the leak is needed tomaintain the operational status in order to provide safety forthe crew. The first indication of a leak in the ISS is the de-pressurization of a module. The leak size can be calculatedby measuring the internal pressure and its depressurizationrate. Based on the extent of leak it is possible to calculatethe “reserve time” left until a crew evacuation is required.Depending on the reserve time operational decisions mustbe made, including: 1) whether or not to perform a leakisolation to patch the leak, or 2) evacuate the ISS. Leaklocalization should be performed first to find the leakingmodule. Then the exact location within the leaking modulefor repair purposes can be determined.

Conventional methods to locate air leaks on the ISS in-clude the sequential module leak isolation process for theUS segment (prior to assembly stage 10A) and the air-flow induction sensor system for the Russian segment. Thesequential module leak isolation process involves havingthe crew close hatches sequentially while monitoring thepressure difference across each hatch. A drawback of thisprocess is very small pressure differences can keep a closedhatch from being open again, which significantly reducesthe reserve time and can pose an immediate risk to the crew.Thus, safety dictates that the hatches be closed in an orderthat will never trap a crew member away from the escapevehicle. This may significantly inhibit the leak isolationprocess if the leaking module is not located within the firstfew hatch closures.

The airflow induction sensor system employs hot-wireanenometers situated in hatchways to measure the air flowdirection and its rate. The hot-wire anenometer operates byair passing across a wire with a current running through itto maintain a constant temperature in the wire. These de-vices are installed at all hatchways of the Russian segments.However, the airflow induction sensor system designed forthe ISS has several limitations for the following reasons.The sensors are not mounted at all hatchways of the USsegment (only at Node-2 and Node-3 of the US segment).Therefore the sequential module isolation process is stillneeded to determine which module leaks in the US seg-ments. Since the sensors are very sensitive to the air circu-lation inside, the venting system and the movement of thecrew must be stopped for several minutes, which may wastetime in an emergency situation. Because these sensors aresituated in hatchways, the location of the leak within thesuspected leaking module cannot be found for repair pur-poses without using other inspection processes (this is alsotrue for the sequential isolation process). Therefore a moreefficient localization system is needed to locate the leaks.

The new method presented in this paper uses the attituderesponse of the ISS caused by the leak reaction force of theair flowing through a perforated hole. The vent thrust canyield a strong reaction torque depending on the size and

location of the leak. A leak hole on the surface of a pres-surized module can be modeled as a short nozzle with theleaking air as the propellant. We assume that the line ofaction of the vent thrust is perpendicular to the cross sec-tion area of the leak hole. This assumption is reasonabledue to the relatively thin skin of each module. Based on thenozzle dynamics, an extended Kalman filter (EKF) algo-rithm is used to estimate the vent thrust magnitude with theinternal pressure measurements. The venting torque is esti-mated by the Unscented Filter (UF) developed by Julier andUhlman.3 The advantages of the UF are: 1) it captures theposterior mean and covariance of a random variable accu-rately to the second-order Taylor series expansion for anynonlinearity by choosing a minimal set of sample pointsand propagating them through the original nonlinear sys-tem, 2) it is derivative-free when the filter equations arealready expressed in discrete forms, i.e. no Jacobian andHessian calculations need to be evaluated for the computa-tion which enable the UF to be applied to any complex dy-namical system and to non-differentiable functions.4 Thevent torque, which is not explicitly modeled in the attitudedynamics, shows up as a residual disturbance torque whenthe spacecraft angular rate measurement undergoes a filter-ing process. In the disturbance torque estimation algorithm,the filter state vector is augmented to include the unknownparameters as additional states, resulting in a total of sixstates, where three states are for the total angular momen-tum of the spacecraft and the remaining three states are forthe 3-axis components of the disturbance torque. But prob-lems arise when the unmodeled dynamics (besides the venttorque) dominate the residual torque.

Among the external disturbances, the aerodynamictorque is known to have large uncertainties in its parametersbut has relatively less effects on the residual disturbancetorque estimation results compared to the uncertainties inthe inertia components of the ISS. Therefore parameter es-timation methods is employed to estimate the six inertiacomponents when we know that there is not a venting leakacting on the spacecraft. The UF algorithm is employed toestimate the inertia in real time. It is shown that the com-plete inertia parameters are unobservable when the spacestation attitude is in its torque equilibrium attitude (TEA),which is the nominal ISS operational attitude. But the in-ertia observability can be strengthened with the presenceof attitude maneuvers. Problems in estimating uncertaininertia and external disturbance torque for the ISS are in-vestigated in several papers such as Refs. 5 and 6. InRef. 7, small sinusoidal probing signals are used to enhancethe observability of the inertia by causing attitude motionabout the TEA. Also in Ref. 6, the estimation algorithmto determine the mass and aerodynamic torque propertiesof the ISS in Low-Earth Orbit (LEO) based on the least-square method has been derived with the use of an indirectadaptive control algorithm to enhance the observability ofthe unknown parameters. This method uses a smoothingmethod to estimate the unmeasured angular acceleration.

The possible locations of the air leak are then calculated

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using the estimated vent torque, vent thrust magnitude,and the actual geometric structure of the pressurized seg-ments. For simplicity, the disturbance torque caused bythe pressure of the impingement of the leaking air plumeon nearby surfaces is neglected. Also, we assume thatthe leak is caused by a single leak hole. There may besingle or multiple leak locations that produce the same atti-tude response. To reduce the number of possible solutions,conventional methods are combined with the new leak lo-calization method. This approach reduces the number ofpossible solutions, so that fewer hatch closures are requiredto uniquely determine the leak location. Advantages of theattitude response method include:

1. No other devices are needed besides pressure gaugesto measure the air pressure, and spacecraft attitude and ratesensors.

2. Relatively fast leak localization can be achieved com-pared to the conventional leak localization method pro-posed for the ISS.

3. The new method not only determines the possibleleaking modules but also determines the possible locationsof the leak hole within those modules. This may be criti-cal to allow for repairs rather than sealing off a module orperforming a station evacuation.

The remainder of paper is organized as follows. First,a summary of the attitude dynamics for the ISS is given.Next, using the isentropic nozzle theory, the vent thrustis calculated using the isentropic and isothermal air de-pressurization models. Then a brief explanation on UF isgiven with the derivations of disturbance torque estimation.Then we describe the steps to locate leak once we know thedisturbance torque and the thrust due to a leak. Finally nu-merical simulations for the leak localization are presentedwith conclusions.

SPACECRAFT ATTITUDE DYNAMICSThe dynamic equations of rotational motion of a rigid

spacecraft in a LEO environment are given by Euler’s equa-tion:

H = −[

J−1 (H − h)]

× H + Ndrag + Ngrav + dvent

(1)whereH is the total angular momentum of the spacecraftsatisfying

H = Jω + h (2)

and J is the inertia matrix,Ndrag is the aerodynamictorque,Ngrav is the gravity gradient torque,h is the angu-lar momentum of the control moment gyroscopes (CMGs),anddvent is the torque due to a vent. Other environmen-tal effects such as solar radiation and Earth’s albedo areneglected. The effects caused by solar arrays rotationsare omitted since they have little effects on the attitudedynamics but note that the resultant aerodynamic torquesproduced by the arrays rotations may be significant.

The gravity-gradient torque for a circular orbiting space-craft is given by

Ngrav = 3n2C3 × J C3 (3)

Module

Wall

Air

PT

PT

P ∗

T ∗

PB

-F vent-F vent

Pressurized Module

Isentropic Nozzle

Fig. 1 Air Flow Through Leak Hole

whereCi is the ith column of the coordinate transforma-tion matrix from the LVLH orbital reference frame to thebody reference frame, andn is the time-varying orbital fre-quency calculated from orbital elements of the spacecraft.

The aerodynamic torqueNdrag is modelled such thatthe drag force and the center of pressure location are func-tions of the attitude of the spacecraft:

Ndrag = −1

2ρav2

aCDS[

ρcp × C1(q)]

(4)

whereva is the magnitude of the atmospheric velocity withrespect to the spacecraft, which can be approximated as thecircular orbital speed. The atmospheric densityρa is cal-culated using Marshall Engineering Thermosphere (MET)model which accounts the seasonal and diurnal heating ef-fects of the Earth’s atmosphere. The drag coefficientCD isassumed to be constant for a given orientation of the space-craft. Also,S is the attitude dependent frontal area andρcp

is the attitude dependent center of pressure location. Theattitude dependent aerodynamic parameters are calculatedwith the method developed in 8, where the reference areaand the center of pressure are calculated for any orientationby defining interpolation functions. The projected area andthe center of pressure for the three orthogonal body refer-ence axes of the ISS are given in 1 for each assembly stage.

The vent torque is modelled by

dvent = rvent × F vent (5)

wherervent is the moment arm of a vent torque from thecenter of mass of the spacecraft to a leak location, andF vent is a vent thrust. The vent torque is unknown andwill be estimated by treating it as a state in the real-timefilter algorithm.

VENT THRUSTA leak hole perforated on the surface of a pressurized

module will behave like a short length nozzle. The dy-namic properties of the air flow through the leak hole areanalyzed using one dimensional isentropic and isothermalnozzle dynamic models. Fig. 1 shows the diagram of theair flow through the leak hole on the pressurized module,whereT ∗ andP ∗ are the temperature and pressure of the

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air in the leak hole, respectively,T andP are the temper-ature and pressure of the inside of the pressurized module,respectively,F vent is the vent thrust, andPB is the backpressure. The mass flow rate in a leak hole is given by9

m = −AP ∗v∗

RT ∗(6)

whereA is the area of the hole,R is the ideal gas constant(287 N-m/Kg-K), andv∗ is the exhaust velocity of the airsatisfying

v∗ =√

γRT ∗ (7)

whereγ is the specific heat ratio, withγ = 1.4 for an idealgas. The mass flow ratem can be expressed as a function ofthe air inside the pressurized module. This is accomplishedby substituting the following expressions into Eq. (6):

P ∗ = P

(

2

γ + 1

)

γ

γ−1

(8a)

T ∗ = T

(

2

γ + 1

)

(8b)

yielding

m = −AP√

γ√RT

(

2

1 + γ

)

1+γ

2(γ−1)

(9)

The actual mass flow rate can be calculated by multiplyingm in Eq. (6) by the discharge coefficientCD. Using thethrust equation the vent thrust magnitude is given by

|F vent| = CDmv∗ + (P ∗ − Pa)A (10)

wherePa is the ambient pressure which is approximatelyzero for the vacuum of space. Substituting Eqs. (6), (7) and(8) into Eq. (10), and simplifying yields

|F vent| = AP (CDγ + 1)

(

2

γ + 1

)

γ

γ−1

(11)

Note that the magnitude of the vent thrust is proportionalto the pressure inside the module and to the area of theleak hole. This expression is very useful since the ventthrust magnitude is a direct function of the internal pressureP , which can be measured by a pressure sensor. For thecalculation of the hole areaA the following approach isused. The indication of an air leak in a pressurized moduleis the depressurization of the air. The air inside the modulefollows the ideal gas law, given by

P =mRT

V(12)

whereV is the volume of the air. Differentiating Eq. (12)with respect to time and usingm from Eq. (9) gives a de-press rate model. Two kinds of depressurization processmodels are used, depending on the temperature character-istics of the air. For an isentropic air model, whereP andT is related by

T = T0

(

P

P0

)

γ−1γ

(13)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 104

1

2

3

4

5

6

7

8

9

10

11x 10

4

Time (sec)

Inte

rnal

Pre

ssur

e (N

/m2 )

Isentropic

Isothermal

Hole Radius : 0.3 (in)

tres

= 9657 tres

= 13101

Fig. 2 Internal Pressure

the depressurization rateP is

P = −k1AP k2 (14a)

k1 =γ√

RT0γ

VP

1−γ

2γ

0

(

2

γ + 1

)

γ+12(γ−1)

CD (14b)

k2 =3γ − 1

2γ(14c)

For an isothermal process,T is treated as a constant inEq. (12). Therefore the depressurization rateP can be de-rived as

P = −k3AP0 (15a)

k3 =

√RT0γ

V

(

2

1 + γ

)

1+γ

2(γ−1)

CD (15b)

where the subscript0 stands for the initial value and,k1, k2

andk3 are constants. Now, we can calculate the hole areaA by measuring the internal pressureP and its depress rateP .

Comparisons between the isentropic and isothermal gasmodel are shown in Figs. 2 and 3, using the ISS assemblyStage 16A with a leak hole radius of0.3 inch. From Fig. 2,the isentropic gas model gives a faster pressure drop in theinternal pressure than the isothermal gas model. Thereforethe reserve timetres, which is a measure of the time it takesfor the current pressureP to reach the minimum habitablepressurePmin ≈ 490 mmHg, is shorter using the isen-tropic gas model than using the isothermal gas model. Thereserve timetres can be obtained by integrating Eq. (14b)for the isentropic process and Eq. (15a) for the isothermalprocess. The reserve time for the isentropic process is

tres =

(

Pmin

P

)

1−γ

2γ − 1

γ−1

2

AV

√RTγ

(

2

γ+1

)

γ+12(γ−1)

CD

(16)

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 104

2

4

6

8

10

12

14

16

18

20

22

Time (sec)

Leak

For

ce (

N)

Isothermal

Isentropic

Hole Radius : 0.3 (in)

Fig. 3 Vent Thrust Magnitude

where the internal temperatureT can be substituted byPfrom Eq. (13). From Fig. 3 the vent thrust magnitude islarger using the isothermal gas model, meaning the isother-mal gas model produces a greater torque than the isen-tropic gas model. If the leak area hole size is small thenthe isothermal model can be used (since the temperaturewill remain fairly constant), otherwise the isentropic modelshould be used.

VENT THRUST ESTIMATIONSince the actual internal pressure measurements are cor-

rupted by noise, the Kalman filter is used to estimate thehole area which is needed to calculate the magnitude ofvent thrust with Eq. (11). The state equations for the de-pressurization process have the following form

x(t) = f [x(t), t] + η(t) (17)

where the statex(t) = [P (t), A(t)]T and

f [x(t), t] =

[

−k1AP k2

0

]

(18)

for an isentropic process model, and

f [x(t), t] =

[

−k3AP

0

]

(19)

for isothermal process model. The vectorη = [η1, η2]T

is the process noise vector, whereη1 andη2 are Gaussianwhite-noise processes with

E {ηi(t)} = 0 (20a)

E {ηi(t)ηj(t′)} = Qiδi,j(t − t′) (20b)

with i, j = 1, 2. The matrixQi has the following form

Q =

[

σ21 00 σ2

2

]

(21)

where the termsσ21 andσ2

2 are the variances ofη1 andη2,respectively. The internal pressure measurement is mod-elled as

zk = hk [x(tk)] + vk, k = 1, 2, . . . , m (22a)

hk [x(tk)] = Pk, k = 1, 2, . . . , m (22b)

wherem is the number of measurements andvk is the mea-surement noise which satisfies a discrete Gaussian white-noise process with

E {vk} = 0 (23a)

E {vkvk′} = Rkδk,k′ (23b)

The propagation of the state satisfies

˙x(t) = f [x(t), t] (24a)

zk = hk [x(tk)] (24b)

wherex(t) = [P (t), A(t)]T is the state estimate vector.The error covariance propagation matrixP satisfies

P(t) = F [x(t), t]P(t) + P(t)F [x(t), t]T

+ Q (25)

whereF [x(t), t] is given by

F [x(t), t] =∂f [x(t), t]

∂x(t)

∣

∣

∣

∣

∣

x=x

(26a)

=

[

−k1k2AP k2−1 −k1Pk2

0 0

]

(26b)

for an isentropic process, and

F [x(t), t] =∂f [x(t), t]

∂x(t)

∣

∣

∣

∣

∣

x=x

(27a)

=

[

−k3A −k1P

0 0

]

(27b)

for an isothermal process.The state estimate and error covariance updates are given

by

x+

k = x−

k + Kk

[

zk − hk(x−

k )]

(28a)

P+

k =[

I − KkHk(x−

k )]

P−

k (28b)

where the superscript (+) stands for the updated value and(−) stands for the a priori value. The Kalman gain matrixis given by

Kk = P−

k Hk(x−

k )T[

Hk(x−

k )P−

k Hk(x−

k )T + Rk

]−1

(29)where Hk(x−) is the measurement sensitivity matrix,given by

Hk(x−

k ) =∂hk(x(tk)

−)

∂x(tk)

∣

∣

∣

∣

∣

x=x

=[

1 0]

(30)

Thus with the use of the internal pressure measurements theEKF algorithm can be used to estimate the leak hole area.Then the magnitude of vent thrust can be calculated by sub-stituting the estimated values ofP andA into Eq. (11).

5 OF 12

UNSCENTED FILTERAn unscented filtering approach is considered here as an

alternative to the EKF for the attitude and angular rate esti-mation of the ISS. The Unscented Filter (UF) has first beenproposed by Julier and Uhlman in 3. Unlike the EKF, theUF captures the posterior mean and covariance of a ran-dom variable accurately to the second-order Taylor seriesexpansion for any nonlinearity by choosing a minimal setof sample points and propagating them through the orig-inal nonlinear system. Also it is derivative-free, i.e. noJacobian and Hessian calculations need to be evaluated forthe computation. Therefore it can be easily applied to anycomplex dynamical system and to non-differentiable func-tions.4 A detailed description of the error performance ofthe UF over EKF can be found in Refs. 3, 4 and 10. Thegeneral formulation of the UF in discrete-time is describedhere.

Let the discrete-time nonlinear system and observationmodel be

xk+1 = f (xk,uk,wk, k) (31a)

yk+1 = h (xk+1,uk+1, k + 1) + vk+1 (31b)

wherexk+1 and yk+1 is an n dimensional state vectorand observation vector respectively,f andh are the non-linear models,uk is a deterministic input. The vectorswk and vk+1 are zero-mean Gaussian process and mea-surement noise with covariancesQk andRk respectively.Using a numerical integration scheme, a continuous-timesystem model can always be expressed in the discrete-timemodel. In the EKF problems arise because the predictionsare approximated simply as functions of the previous stateestimates:

x−

k+1= E [f (xk,uk, k)] ≈ f (xk,uk, k) (32)

yk+1 = E [h (xk,uk, k)] ≈ h (xk,uk, k) (33)

But if the estimated state is nearby the true value, then thefilter usually has a good convergence.

The UF state and error covariance updates are given as

x+

k = x−

k + Kkυk (34)

υk = yk − y−

k = yk − h(

x−

k ,uk, k)

(35)

P+

k = P−

k − KkP υυk KT

k (36)

Kk = Pxyk (P υυ

k )−1 (37)

whereυk is the innovation andP υυk is the covariance of

υk. The filter gain isKk andPxyk is the cross-correlation

matrix betweenx−

k andy−

k . One of the way to deal with theprocess noise is to augment the covariance matrix with10

P ak =

[

P+

k P xwk

(P xwk )T QK

]

(38)

whereP xwk is the correlation between the state and the pro-

cess noise. The assumption is made here that the measure-ment noisev is purely addictive unlike the process noise.

The set of2L(= 4n) σ points are computed as follows:

σk (i) = ±√

(L + λ) [P ak ]

iwherei = 1, 2, . . . , L

(39)whereλ is the weighting factor which scales the distribu-tion of the points. The vector

√

(L + λ) [P ak ]

irepresents

ith column of the matrix square-root√

(L + λ) [P ak ]. The

matrix square-root can be calculated directly by a lower tri-angular Cholesky factorization method.

Theseσk (i) points translate the meanxak as

χ (0) = xak, χ (i) = xa

k + σk (i) (40)

wherexak is an augmented state defined as

xak =

[

xk

wk

]

, xak =

[

xk

0n

]

(41)

The transformed set ofχ points are propagated tok +1 foreach of the2L + 1 points by

χk+1 (0) = f (χk (0) ,uk, k) , χk+1 (i) = f (χk (i) ,uk, k)(42)

The predicted mean is

x−

k+1=

1

L + λ

{

λχxk+1 (0) +

1

2

2L∑

i=1

χxk+1 (i)

}

(43)

whereχx is a vector of the firstn elements ofχa. Thepredicted covariance is

P−

k+1=

1

L + λ

{

λ[

χxk+1 (0) − x−

k+1

] [

χxk+1 (0) − x−

k+1

]T

+1

2

2L∑

i=1

[

χxk+1 (i) − x−

k+1

] [

χxk+1 (i) − x−

k+1

]T

}

(44)

The predicted observation is calculated as

y−

k+1=

1

L + λ

{

λγk+1 (0) +1

2

2L∑

i=1

γk+1 (i)

}

(45)

where

γk+1 (i) = h(

χxk+1 (i) ,uk+1, k + 1

)

(46)

The output covariance is given by

Pyyk+1

=1

L + λ

{

λ[

γk+1 (0) − y−

k+1

] [

γk+1 (0) − y−

k+1

]T

+1

2

2L∑

i=1

[

γk+1 (i) − y−

k+1

] [

γk+1 (i) − y−

k+1

]T

}

(47)

then the innovation covariance is given by

P vvk+1 = P

yyk+1

+ Rk+1 (48)

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Finally the cross correlation matrix is

Pxyk+1

=1

L + λ

{

λ[

χxk+1 (0) − x−

k+1

] [

γk+1 (0) − y−

k+1

]T

+1

2

2L∑

i=1

[

χxk+1 (i) − x−

k+1

] [

γk+1 (i) − y−

k+1

]T

}

(49)

The filter gain, the state and error covariance update is thencomputed using Eqs. (34) and (37). The weightλ can bechosen asλ = 3 − L if the statex is assumed to belongto the Gaussian distribution.11 Although λ can be eitherpositive or negative, the negative values may lead to a non-positive semi-definite covariance matrix.

The UF is used in attitude, vent torque, and inertia ma-trix estimation in our leak localization algorithm. For theattitude estimation, the Unscented Quaternion Estimator(USQUE) developed by Crassidis and Markley has beenimplemented. For further details on USQUE, refer Ref. 12.

DISTURBANCE TORQUE ESTIMATIONThe vent torque, which is not explicitly modeled in

the attitude dynamics, shows up as a residual disturbancetorque when the spacecraft angular rate measurement un-dergoes a filtering process. In the disturbance torque es-timation algorithm, the filter state vector is augmented toinclude the unknown parameters as additional states, re-sulting in a total of 6 filter states, where 3 states are for theangular rate or angular momentum of the spacecraft andthe rest 3 states are for the 3-axis components of the dis-turbance torque. Note that the attitude quaternion, which isneeded to determine the gravity-gradient and aerodynamictorque in the disturbance torque estimation algorithm, is es-timated separately by the USQUE method described in theprevious section. In this section the disturbance estimationalgorithm using the UF approach is shown.

The state system model for the torque estimation filterwith x = [H, dvent]

T can be expressed as[

H

dvent

]

=

[

fH(H,dvent)fd(H,dvent)

]

+

[

ηH

ηd

]

=

[

−J−1 (H − h) × H + L + dvent

03×1

]

+

[

ηH

ηd

]

(50)

wheredvent is the vent disturbance torque,ηH and ηd

are zero-mean Gaussian process noises which correspondroughly to the possible range of the disturbance variations.The quantityL is the external disturbance torque vectorwhich can be expressed as

L = Ndrag + Ngrav (51)

The quantityL is treated as a deterministic input in thefilter equations. Also the CMG control inputh should below pass filtered because of the presence of high-frequencynoise when measuring the CMG wheel speed. Note that the

2.071 2.0715 2.072 2.0725 2.073 2.0735 2.074

x 104

−50

0

50

100

150

2.071 2.0715 2.072 2.0725 2.073 2.0735 2.074

x 104

−100

0

100

2.071 2.0715 2.072 2.0725 2.073 2.0735 2.074

x 104

−200

0

200

400

600

time (sec)

d1

(Nm

)d2

(Nm

)d3

(Nm

)

Fig. 4 Vent Torque Estimate Using UF with True Value

0 1 2 3 4 5 6−2

−1

0

1

2

0 1 2 3 4 5 6−2

−1

0

1

2

0 1 2 3 4 5 6−2

−1

0

1

2

time (hr)

err d

1(N

m)

err d

2(N

m)

err d

3(N

m)

Fig. 5 Disturbance Torque Estimation Error Using UF

vent torquedvent is treated as a random walk process. Thegyroscope output measurement model is

y = ω + η1

= J−1 (H − h) + η1 (52)

Unlike the EKF, the UF approach does not need anyderivation of the Jacobian and the sensitivity matrices. Forthe UF we need only the original nonlinear equation andmeasurement model. In the disturbance estimation algo-rithm, the UF approach may be especially more robust thanthe EKF because the initial conditions for the angular ratecomponents may be fairly accurate within the uncertaintiesof the gyroscope, but the initial guess of the disturbancetorque, which is not measured, may be far from the truevalue.

Numerical simulations for the UF cases are performedwith the angular rate noise standard deviation of2.3 ×10−4 ◦/sec (σ1 = 4 × 10−6 rad/sec) and the samplingfrequency of 1 Hz for the ISS assembly stage UF1. It is as-

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sumed that the spacecraft attitude is maintaining the TEAwhen suddenly after 5.7556 hr (20720 sec) a vent torqueof 66.07 Nm is applied in each body axis of the spacecraft.The disturbance torque estimate results after the vent areshown in Fig. 4, where the dashed lines correspond to thetrue values. We can see that the vent torque estimates con-verge to the true values around 10 seconds after the leak.When an air leak occurs, the state covariance of the fil-ter is reset to a large value to incorporate the variation ofthe disturbance torque at the instant when leak occurs (re-member that we know when leak occurs by sensing the airdepress inside the crew cabin). In this way the filter con-verges much faster than that of the filter algorithm withouta covariance reset. The estimation errors for each compo-nent of the disturbance torque are shown in Fig. 5 with their3σ-bound lines.

INERTIA ESTIMATIONFor the ISS, the uncertainty in the aerodynamic torque

may affect the vent torque estimation results if they havethe same order of magnitude with the torque due to a leak.But the major uncertainty in the residual torque estimationis likely caused by the inaccurate ISS inertia mass com-ponents. For the ISS, the inertia of each configuration ispre-calculated on ground with CAD tools. But these valuesmay not be precise since the ISS is made up of multiplecomplex rigid bodies interconnected to each other and un-dergoes several configuration changes during its lifetime.

Therefore, online parameter estimation method may beemployed to estimate these slowly changing inertia in real-time when we know that there is no venting leak acting onthe spacecraft. But the parameter estimation performancedepends heavily on the observability of the parameters ofinterest. Usually in the parameter estimation problem, thestate vector is extended by adjoining it with the vector ofunknown parameters, as we have done for the vent torqueestimation algorithm. In this section, we use the least-square approach to analyze the relative observability of theISS inertia components.

When the ISS attitude is near the LVLH, the inertia ma-trix are unobservable even though there are some slight atti-tude variations due to the time varying aerodynamic torque.Assuming that the aerodynamic parameters are known, theinertia matrix observability in an ideal LVLH fixed modecan be shown from the following equations:

J23 =1

4n2(τaero 1 − u1)

J13 =1

3n2(u2 − τaero 2)

J12 =1

n2(u3 − τaero 3) (53)

where the constant angular rateω = [0 − n 0]T andthe constant attitude quaternionq = [0 0 0 1]T are sub-stituted in the rotational Euler equations of motion. Thequantitiesτaero i andui are theith component of the aero-dynamic torque and the control torque input, respectively,

andJij is the ijth inertia matrix element. The spacecraftis assumed to be rotating in an Earth-pointing mode witha constant attitude angular raten = 0.0011 rad/sec. Wecan see from Eqs. (53) that among the six inertia compo-nents, only the products of inertia (J23, J13 andJ12) showup due to the presence of the gravity-gradient torque. Butnote that the control inputu and the aerodynamic torqueτ aero have small values with the same order of magnitude.Therefore, exact knowledge of the aerodynamic and con-trol input torque are needed to directly calculate the productof inertias, which is not feasible in a real world.

A numerical test is done with the batch least-squaremethod to check the observability conditions in the LVLHfixed attitude mode. The assumptions are: 1) perfect mea-surements of the attitude, angular rate, control input and theangular acceleration are available, 2) perfect knowledge ofthe aerodynamic torque, 3) no other disturbances besidesaerodynamic and gravity-gradient torque are present. Thesolution of the least-square method for the estimation of theinertia matrix is as follows:

x = (HT H)−1HT y (54)

wherex is a 6-dimensional vector containing the elementsof the inertia matrix as

x = J = [J11 J22 J33 J23 J13 J12]T (55)

and theH and they are quantities known from the mea-surements and the control inputs. Note that a linearparametrization of the equations of motion is needed to usethe batch least-square method. The Euler equations can belinearly parameterized with respect to the unknown inertiacomponents as

−u + τ aero = Jω + ω × Jω − 3n2C3 × JC3

= [D1(ω) + D3(ω) − 3n2D3(C3)]J

y = HJ (56)

where the matricesD1 andD3 are defined as in 6.The quantityHT H should be strictly positive definite

since its inverse appears in Eq. (54) to solve the unknownparametersx. In practice, we requireHT H to be well-conditioned, a useful measure of the condition of a matrixis the condition number. The condition number varies from1 for an orthogonal matrix to infinity for a singular matrix.From a numerical simulation, when all six components ofinertia matrix are solved using the Eq. (54), the conditionnumber of theHT H is 1.7 × 1010 resulting in the diver-gence of the solution. The relative observability among theinertia components is analyzed using the eigenvalue andeigenvector decomposition of theHT H matrix, which isshown in Fig. 6. In this figure we can see the relative ob-servability of the inertia components. For example,J11 isthe maximum component of the eigenvector which corre-sponds to the eigenvalue around10−15. As expected thethree products of inertia, which have their eigenvalues near10−6 are the most observable components among the el-ements, whereas the three moments of inertia have their

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−18 −16 −14 −12 −10 −8 −6 −40

1

2

3

4

5

6

7

Eigenvalues inLog10(·)

Com

pone

ntIn

dex

J11

J22

J33

J23

J13

J12

Fig. 6 Relative Observability of Inertia

magnitude near10−15 which is 10−9 smaller than thoseof the products of inertia. A simulation has been done toestimate the product of inertia with a batch least-squaremethod, and the results are shown in Fig. 7 (where theresults are calculated at regular instant of time with thecumulative measurements). The three components con-

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−10

−5

0

5

x 104

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−5

0

5

10x 10

6

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−10

−5

0

5x 10

5

time (orbit)

J23

(kg

m2

J13

(kg

m2

J12

(kg

m2

Fig. 7 Inertia Product Estimates Using Least-Square Method

verge very fast within an orbit to its true values as expected.The corresponding condition number is 16 which is muchsmaller than the previous simulation case revealing that theHT H is now a well-conditioned matrix. But note that thepresence of noise in the measurements makes the productsof inertia unobservable. For the real-time estimation of theinertia, the UF approach is used because of its robustnessin the presence of large initial state uncertainty. Also, theinertia estimation should be performed only when an atti-tude maneuver is present to enhance the observability ofthe parameters.

LEAK LOCALIZATIONOnce a vent torquedvent is estimated by the real-time

filter, the next step involves determining the position vectorrvent, which is the moment arm of the vent torque satisfy-ing

dvent = rvent × F vent (57)

In the above equation, the vent torquedvent and the magni-tude ofF vent are known by the estimation algorithms. Theoverall steps for locating a leak on the ISS are as follows:

1. Model the 3 dimensional geometric surfaces of thepressurized parts of the spacecraft.2. Estimate the vent torque and magnitude of the ventthrust.3. Slice the 3-D surfaces of the pressurized modules witha plane perpendicular to the direction of the vent torque sothat this plane comprises the center of mass of the space-craft. From the fundamental definition of torque, a torqueabout the center of mass of a rigid body is perpendicularto the plane comprising the vectorsrvent andF vent. So,rvent, F vent and the center of mass are all in the sameplane normal to the direction of the vent torque. Denotethis plane byτ . The intersection between the planeτ andthe surface of the spacecraft produces contours.4. With the assumption that the vent thrust is normal tothe tangent plane of the partial section on the ISS surfacewhere the leak occurs, calculate the gradient vectors (direc-tion normal vectors) of the points that make up the slicedcontours obtained in Step 3.5. Multiply the magnitude of the vent thrust estimated inStep 1 with all gradient vectors calculated in Step 4.6. Since the position and gradient vectors of all the pointsmaking the sliced contours are known, calculate the result-ing torque at each point on the contours.7. From the torques obtained for each point in Step 6, selectthe torques that are closest to the estimated torque (withinan error bound) and check their points on the contours.

The actual geometric structure of the station eliminatesmany of the possible solutions; however, multiple solutionsmay still exist. In this case further assumptions can bemade, such as the probability of impacts by the debris orsmall meteorites is low on the aft and nadir facing surfacessince these surfaces are shaded by other structures. Also,the leak localization method based on the attitude responsemay be combined with the conventional leak localizationmethods. For example, if the solution shows two leaks sit-uated at two different modules then only one hatch closurebetween any of these modules is needed to check which oneof the two modules leaks. Furthermore, visual inspectionsby the crew may narrow the possible leak solutions.

NUMERICAL SIMULATIONA user-friendly design tool coded entirely in MATLAB

has been developed to estimate a leak location under var-ious conditions. The tool supports several ISS assemblystages from 11A to UF-7 but it may need to be modifieddue to the uncertainties in the future of the ISS program.The 3-D surface models of the pressurized segment of theISS Stage have been developed based on the data providedin Ref. 1. Figures 8 and 9 show the main Graphical UserInterfaces (GUIs) of the tool. Users can input the orbit,the mass and the aerodynamic parameters of the ISS, andchoose a simulated leak location with the GUI shown inFig. 8. The resulting leak locations after the leak localiza-

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Fig. 8 GUI for Simulation of the Leak Localization Algorithm

Fig. 9 GUI for Leak Localization Results

tion estimation process will be shown on the GUI shown inFig. 9.

For the simulation, the ISS assembly Stage 16A is con-sidered (see Fig. 10). The isentropic depressurization pro-cess of the air inside the ISS is assumed. The mass andaerodynamic properties of the ISS are provided in Ref. 1.The inertiaJ is given by

J =

127908568 3141229 77091083141229 107362480 13452797709108 1345279 200432320

(kg m)2

(58)

The centers of pressure areρcpx = [0,−0.355,−0.927]T

m, ρcpy = [−7.94, 0,−1.1]T m and ρcpz =

[1.12, 0.247, 0]T m in the Space Station Analysis Coor-dinate System (SSACS) with respect to the center of mass.The componentsx, y andz represent the three orthogonalaxes of the ISS body fixed frame.1 The reference projectedareas areSx = 967 m2, Sy = 799 m2 andSz = 3525 m2.

Fig. 10 ISS Assembly Stage 16A

0 10 20 30 40 50 60 70 80 90 1003

4

5

6

7x 10

4

Pre

ssur

e (P

asca

l)

0 10 20 30 40 50 60 70 80 90 1000

0.5

1x 10

−3

Are

a (m

2 )

0 10 20 30 40 50 60 70 80 90 1000

50

100

150

200

|FLe

ak| (

N)

Time (sec)

Fig. 11 Vent Thrust Magnitude and Hole Area Estimate

The Global Positioning System (GPS) attitude-sensormeasurement-error standard deviation is given byσq = 0.5deg, and the ring-laser gyro sensor measurement-error stan-dard deviation is given byσω = 4 × 10−6 deg/sec.13 Themeasurement-error standard deviation of the internal pres-sure is given byσ = 0.1 mmHg. For the depressurizationof the air inside, the initial internal temperature and pres-sure are set toT0 = 21o C andP0 = 1 atm, respectively.The back pressure is assumed to bePB = 0 atm, andthe volume of the entire pressurized system for ISS 16Ais V = 867.2 m3. Finally, an inertia uncertainty of 3% isadded to the trueJ .

Simulations are done for 100 seconds from the start ofthe leak. Figure 11 shows the estimate of the leak holearea using the Kalman filter algorithm. The true leak holeareaA is 1.8241 × 10−4 m2. As seen from this figure theKalman filter accurately estimates the leak hole area. Thevent thrust magnitude is then computed with the internalpressure measurement and the estimate of the hole area.

For the first simulation, a leak is assumed on a moduleshown in Fig. 12. The sliced planeτ with contours in 3-D

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Fig. 12 Slicing 3-D Surface Model with Plane τ

−20−15

−10−5

05−10

0

10

20

30

40

−101

2

68

12

1919

21

21

22

XΓ (m)

YΓ (m)

Γ Plane

Fig. 13 Sliced Plane τ with Contours in 3-D

−40 −30 −20 −10 0 10 20

−10

−5

0

5

10

15

20

25

30

35

40

2

68

12

1919

21

21

22

xtau

ytau

tauplane

Fig. 14 Contours on Plane τ with Possible Leak

−20 −10 0 10 20 30−30

−25

−20

−15

−10

−5

0

5

10

15

1

1

3

3

4

4

6

10

1011

1111

1313

141416

16

1818

19

1920

20

21

21

21

Xτ (m)

Yτ (

m)

P1

P3 P

2

PLANE τ

Fig. 15 Contours on Plane τ with 3 Possible Leaks

is shown in Fig. 13. Using the leak localization approacha single leak has been determined for this simulated case,depicted in Fig. 14. The estimated position is marked witha ◦, the true position of a leak is marked with a∗ for com-parison, and the center of mass is marked with a⋆ on theplaneτ . Slicing of the 3-D surface is performed at the endof the simulation (t = 100 sec). If no errors are presentin the assumed model and if the assumptions made so farare perfectly satisfied, then the closest torque yielding thepoint to the estimated vent torque is the true leak point.But because of sensor inaccuracies and modelling errors inthe inertia, the estimated vent torque may deviate from thetrue value. Therefore, an upper error-bound should be setwhen selecting points that yield the torque closest to theestimated vent torque. For the case shown in Fig. 14, weconclude that the leak occurs on the contour line labelled8, which corresponds to the Kibo JEM pressurized module.In this simulated case, the leak location is well estimatedusing the new localization method.

Another simulation has been done where multiple loca-tions may result from the given estimated vent torque. Inthis case the estimated leak locations are spread over sev-eral modules, as shown in Fig. 15. The locationsP1, P2

andP3 are possible leak candidates (the true leak point issituated nearP1). But sinceP1 andP2 are on the samemodule, a crew person only needs to close one hatch be-tween the module labelled 20 and the module labelled 19to verify which one of the two modules has a leak. This isaccomplished by measuring the internal pressure drop rateor using visual inspections of the estimated leak points. Ifthe leak hole is due to space debris or small meteorite punc-tures, then the aft and nadir facing surfaces of the ISS havelittle possibility to be impacted. This is also true for loca-tions where regions are protected by other structures, as isthe case for pointP3. Therefore this point is not a likelycandidate for the leak.

Initial results indicate that the leak localization methodmay be sensitive to modelling errors, such as the spacecraftmass properties and aerodynamic parameters. Also the ef-fect of the disturbance torque caused by the pressure of theimpingement of the leaking air plume on nearby surfaces

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may be a critical source of disturbance when a leak occurs.Because of inherent complexities, analyzing these effectsmay be difficult. Experimental validation of the algorithmhas been treated in Ref. 14.

CONCLUSIONSIn this paper a new leak localization method using the

attitude response is developed for the ISS. The reactionthrust arising from a vent due to air leak are calculated us-ing the isentropic nozzle theory. Also, the isentropic andisothermal depressurization models have been consideredto describe the depressurization process of the pressurizedmodule. Based on these models, the vent thrust magni-tude is estimated by employing the KF. The UF approachis used for the purpose of estimating the attitude and resid-ual disturbance torque. The UF approach is preferred overthe EKF since the expected error is lower and it avoids thederivation of complicated Jacobian matrices.

In the ISS, an incorrect inertia may be the primary sourceof uncertainty in estimating the vent torque. Also, unlikethe deterministic gravity-gradient, the precise determina-tion of the aerodynamic torque is very difficult due to thelack of the knowledge of the drag mechanism in rarefiedatmosphere activities. But since the upper bound of theaerodynamic torque is known, the vent torque which ismuch larger than the aerodynamic torque has no much ef-fect on the vent torque estimation results. It has been shownwith the batch least-square analysis that the inertia matrixis unobservable when the ISS is near the LVLH. Therefore,to enhance the observability of the unknown parameters,an appropriate attitude maneuver should be performed. Forthe real-time inertia parameter estimation, the robustnessin the presence of large initial state uncertainty and nonecessities of the Jacobian derivations may make the UFapproach attractive.

The actual geometric structure of the station eliminatesmany of the possible solutions; however, multiple solutionsmay still exist. In this case further assumptions should bemade, such as the probability of impacts by the debris orsmall meteorites is low on the aft, nadir facing surfacesand some parts of the surfaces which is not likely to haveleak. Also, the leak localization method based on the atti-tude response may be combined with the conventional leaklocalization methods. For example, if the solution showstwo leaks situated at two different modules then only onehatch closure between any of these modules is needed tocheck which one of the two modules leaks. Furthermore,visual inspections by the crew may narrow the possible leaksolutions.

Numerical results showed that the proposed leak local-ization method determines the location of the leak rapidlyand precisely. Furthermore, actual test data from a depres-surization of the Space Shuttle airlock indicates that theproposed method has the potential to accurately estimatethe leak hole size and venting force magnitude.

The advantages of this localization method are: no otherdevices are needed besides pressure gauges, spacecraft at-

titude and rate sensors, and relatively fast leak localizationcan be achieved compared to the conventional leak local-ization method proposed for the ISS. Also this localizationmethod not only determines the possible leaking modulesbut also the possible locations of a leak hole within the sur-faces of the modules, which may be critical to allow forrepairs rather than sealing off the module or performing astation evacuation.

ACKNOWLEDGEMENTSThis work has been supported by a grant from United

Space Alliance (contract number C00-00109). This supportis greatly appreciated.

References1International Space Station On-Orbit Assembly, Modeling, and

Mass Properties Databook: Design Analysis Cycle Assembly Sequence,JSC-26557, Revision J, Lockheed Martin, Houston, TX, 1999.

2Protecting the Space Station from Meteoroids and Orbital Debris,National Research Council, Washington, DC, 1997.

3Julier, S. J., Uhlmann, J. K., and Durrant-Whyte, H. F., “A New Ap-proach for Filtering Nonlinear Systems,”American Control Conference,pp. 1628-1632, Seattle, WA, 1995.

4Wan, E. and van der Merwe, R., “The Unscented Kalman Filter forNonlinear Estimation,”Proceedings of IEEE Symposium 2000, pp. 153-158, Lake Louise, Alberta, Canada, Oct. 2000.

5Bishop, R. H., Paynter, S. J., and Sunkel, J. W., “Adaptive Control ofSpace Station During Nominal Operations with CMGs,”IEEE Conferenceon Decision and Control, Brighton, England, Dec. 11-13 1991.

6Carter, M. T.,Real-Time Analysis of Mass Properties, Dissertation,Texas A&M University, College Station, February 1999.

7Paynter, S. J.,Adaptive Control of the Space Station, M.s. thesis,University of Texas, Austin, December 1992.

8Carter, M. T. and Vadali, S. R., “Parameter Identification fortheInternational Space Station Using Nonlinear Momentum ManagementControl,” AIAA-97-3524, August 1997.

9Sonntag, R. E., Borgnakke, C., and Wylen, G. J. V.,Fundamentalsof Thermodynamics, John Wiley & Sons, Inc., New York, 5th ed., 1998.

10Crassidis, J. L. and Junkins, J. L.,Optimal Estimation of DynamicalSystems, CRC Press, Boca Raton, FL, 2004.

11Julier, S. and Uhlmann, J., “A New Extension of the Kalman Filterto Nonlinear Systems,”Int. Symp. Aerospace/Defense Sensing, Simul. andControls, Orlando, FL, 1997.

12Crassidis, J. L. and Markley, F. L., “Unscented Filtering for Space-craft Attitude Estimation,”Journal of Guidance, Control, and Dynamics,Vol. 26, No. 4, July-August 2003, pp. 536–537.

13United States Control Module Guidance, Navigation, and ControlSubsystem Design Concept, NASA MSFC-3677, Huntsville, AL, March1997.

14Kim, J. W., Crassidis, J. L., Vadali, S. R., and Dershowitz, A.L., “In-ternational Space Station Leak Localization Using Attitude DisturbanceEstimation,” Vol. 7, Aerospace Conference, Proceedings. 2003 IEEE,March 8-15 2003.

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